The effect of ideology on policy outcomes in proportional representation systems

doi:10.1016/j.mathsocsci.2011.05.001

Mathematical Social Sciences

Volume 62, Issue 2, September 2011, Pages 87-90

https://doi.org/10.1016/j.mathsocsci.2011.05.001 Get rights and content

Abstract

In this paper we propose a model in which there are ideological and strategic voters who vote under proportional rule. We prove that the behavior of ideological voters matters for the determination of the outcome. We show that a subset of strategic voters partially counteracts the votes of the ideological voters.

Introduction

Electorates are typically constituted by two types of voters, those who are committed to vote for a specific party — its hard-core supporters, and those who vote in a more strategic fashion to get their way as regards the policies that are going to be implemented. The result of a general election can be expected to be influenced by the interplay of the behavior of these two types of voters. The present paper addresses such an interplay in the context of an election held with a proportional system.

Specifically, we study a society composed of policy motivated strategic citizens and ideological citizens, who vote for one of a finite number of parties by proportional rule. Given the electoral result, the policy outcome is a linear combination of the position of each party, weighed by the share of votes a party gets in the election. We ask, first, if the ideological voters’ behavior affects the final outcome; second, how strategic voters respond to that.

We prove that the behavior of ideological voters matters for the outcome. In particular, we show that the policy will, in general, be different with respect to the case where all voters act strategically, even with an arbitrarily small number of ideological voters. Concerning the second question, we show how some strategic voters change their voting behavior to, at least partially, counteract the ideological citizens’ vote. Strategic voters will vote in accord with a cutpoint outcome: in equilibrium, any strategic voter on the right of the cutpoint votes for the rightmost party and any strategic voter on its left votes for the leftmost party. The intuition is the following. Given the ideological voting behavior, strategic voters misrepresent their preferences by voting for the extremist parties in order to drag the policy outcome toward their preferred policy.

The model extends to an environment with ideological voters, the analysis of De Sinopoli and Iannantuoni (2007), who study strategic voting under proportional rule and find that essentially only a two-party equilibrium exists, in which voters vote only for the two extremist parties. The voting literature (Shepsle, 1991, Cox, 1997, Persson and Tabellini, 2000) has dealt with models in which either all voters are strategic or all are ideological. An analysis of the more realistic case in which both types coexist is missing. This paper also reconciles the two-party equilibrium result in De Sinopoli and Iannantuoni (2007), with the view that proportional systems should lead to multipartyism (see Cox, 1997).

The rest of the paper is organized as follows. In Section 2 we describe the model; we present an example in Section 3; we analyze the pure strategy equilibria, and, then, the mixed strategy ones in Section 4; we go back to the example in Section 5; Section 6 concludes.

Section snippets

The model

Policy space. The policy space $X$ is a closed interval of the real line. Without loss of generality, we assume $X = [0, 1]$ .

Parties. There is an exogenously given set of parties $M = {1, \dots, k, \dots, m}$ , with $m \geq 2$ , indexed by $k$ . Each party $k$ is characterized by a policy $ζ_{k} \in [0, 1]$ . In order to simplify the notation, in the following we will denote $L$ the leftmost party and $R$ the rightmost (i.e., $L = arg {min}_{k \in M} ζ_{k}, R = arg {max}_{k \in M} ζ_{k}$ ).

Voters. There is a finite set of voters $N = {1, \dots, i, \dots, n}$ . Each voter $i$ , characterized by a

Example 1

Before moving on to the solution, an illustrative example is in order.¹

The equilibrium

We start the analysis of strategic voters’ behavior by first focusing on the case when players only play pure strategies. We start with an intuitive but key result for rational voters’ behavior: in every pure strategy equilibrium strategic voters vote for one of the two extremist parties, except for a neighborhood whose length is inversely proportional to the total number of players.

Proposition 1

Let $s$ be a pure strategy equilibrium of a game $Γ$ with $n$ voters:

$(α)$
$\forall i \in N^{ρ}$ , if $θ_{i} \leq X (s) - \frac{1}{n} (ζ_{R} - ζ_{L})$ then $s_{i} = L$ ,
$(β)$
$\forall i \in N^{ρ}$ , if $θ_{i} \geq X$

Example 1 (continued)

We compute the cutpoint for the example provided above. The effect pertaining to strategic voters is $(0) {\bar{H}}^{ρ} ({\tilde{θ}}_{ρ}^{Γ}) + (1) (1 - {\bar{H}}^{ρ} ({\tilde{θ}}_{ρ}^{Γ})),$ while the “fixed effect” of ideological voters is $\sum_{k = 1}^{3} \frac{n_{k}^{ι}}{n^{ι}} ζ_{k} = \frac{40}{100} (0) + \frac{20}{100} (0.5) + \frac{40}{100} (1) = 0.5 .$ Since there is the same number of strategic and ideological voters, the weights for the two effects are equal and the cutpoint is ${\tilde{θ}}_{ρ}^{Γ} = \frac{1}{2} (1 - {\bar{H}}^{ρ} ({\tilde{θ}}_{ρ}^{Γ})) + \frac{1}{2} (0.5) = 0.45 .$ Hence, only the voters with bliss point in 0.9 will vote for $R$ , while all the others for $L$ in any Nash

Conclusion

We have provided a model in which there are policy motivated strategic voters who take their voting decision maximizing their utilities, and ideological voters, who simply cast their ballot in favor of the party whose policy is closest to their preferred one. The main question has been whether ideological voting behavior really matters. The answer has been affirmative. We have proved that there is basically a unique Nash equilibrium characterized by a cutpoint outcome such that any strategic

Acknowledgments

The authors acknowledge research funding from the Spanish MEC, Grant SEJ2006-11665-C02-0. Giovanna acknowledges research grant PRIN of the Italian MIUR “Political Economics: Theory and Evidence”.

References (5)

G. Cox
Making Votes Count
(1997)
F. De Sinopoli et al.
A spatial voting model where proportional rule leads to two-party equilibria
International Journal of Game Theory
(2007)

There are more references available in the full text version of this article.

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^☆: An earlier version of this paper was circulated under the title “On the effect of ideology in proportional representation systems” (working paper n. 160, Economics Dep. Università di Milano, Bicocca).

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